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THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING
Lance Fortnow Georgia Institute of Technology
THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance - - PowerPoint PPT Presentation
THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance - - PowerPoint PPT Presentation
THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING Lance Fortnow Georgia Institute of Technology Gdel Letter to von Neumann (1956) One can obviously easily construct a Turing machine, which for every formula F in first order
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Gödel Letter to von Neumann (1956)
One can obviously easily construct a Turing machine, which for every formula F in first order predicate logic and every natural number n, allows one to decide if there is a proof of F of length n (length = number of symbols). Let ψ(F,n) be the number of steps the machine requires for this and let φ(n) = maxF ψ(F,n). The question is how fast φ(n) grows for an optimal machine. One can show that φ(n) ≥ k n. If there really were a machine with φ(n) k n (or even k n2), this would have consequences of the greatest importance. Namely, it would
- bviously mean that in spite of the undecidability of the
Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.
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■ Finite Alphabet ■ String is a sequence of characters – Can encode objects like logical formula. ■ Language is a set of strings. – Example: Set of tautologies
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Turing machine M computes a language L if for all strings x – If x is in L then M(x) ends in an accept state – If x is not in L then M(X) ends in a reject state
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P and NP
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NP-completeness
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Clique
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Traveling Salesman
13,509 cities with population at least 500
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Map Coloring
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DNA Sequencing
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Sudoku
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Sudoku
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Rubik’s Cube
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NP-Complete
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Proving P NP
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THE P V NP PROBLEM IN THE ERA OF BIG DATA AND FAST COMPUTING
Lance Fortnow Georgia Institute of Technology
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Brute Force Heuristics Approximation Solve a Different Problem Give Up If P NP: Need to Solve Hard Problems
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1971
3000 Transistors
2005
230 Million Transistors
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SAT Solvers
Can solve satisfiability problems of hundreds of variables. Does really well on problems with tens
- f thousands to millions of variables.
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Linear Programming
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Integer Programming
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Mixed Integer Programming
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Is P v NP Relevant Today?
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Cryptography
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DOG
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MUFFIN
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William of Ockham, English Franciscan Friar Occam’s Razor (14th Century)
Entia non sunt multiplicanda praeter necessitatem
Occam’s Razor
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William of Ockham English Franciscan Friar Occam’s Razor (14th Century)
Entities must not be multiplied beyond necessity The simplest explanation is usually the best.
Occam’s Razor
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Data consists of a random example of some structure.
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00010101000100000101
Structure: Every odd bit is a zero Random: Even bits
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Kolmogorov Complexity
K(x) is the smallest program that generates x x is random if K(x) is at least |x|
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Kolmogorov Structure Function Minimum Description Length
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Kolmogorov Structure Function Minimum Description Length
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If P = NP
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